Very recently, He-Long [1]} investigated the existence and concentrating behavior of the least energy solutions with given \(L^{2}\) -norm for (REF ) under \(s\in (0,1)\) , \(m(x)\equiv 1\)  and \(V(x)\)  being bounded. Luo-Zhang [2]} studied the existence and nonexistence of normalized solutions for (REF ) with \(V(x)\equiv 0\)  and combined nonlinearities, i.e., \(\mu |u|^{q-2}u+|u|^{p-2}u\) , \(2<q<p<2^{*}_{s}=\frac{2N}{N-2s}\) . Motivated by the mentioned works, it seems natural to ask how about the least energy solutions with prescribed \(L^{2}\) - norm for (REF ) in more general case: \(s\in (0,1)\) , \(V(x)\)  being a bounded function and \(m(x)\lnot \equiv 1\) ? The aim of this paper is to give a detailed answer to this question. However, different from the above mentioned papers [3]}, [1]}, [5]}, in our case we should be confront with a series difficulties simultaneously, such as the fractional Laplace operator \((-\Delta )^{s}\)  is nonlocal which makes the energy estimates more difficult, the potential \(m(x)\lnot \equiv 1\)  is involved in the nonlinear term which makes the concentration analysis more complicated, and the potential \(V(x)\)  is bounded (not coercive) which leads to the lack of compactness for Sobolev embedding in \(\mathbb {R}^N\) , etc.
